Finite Group Databasebeta

← #34103471

Subgroups — #34103471

Order 20096.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14578954C211yesmaximal, characteristic
2C2×#19097754C212yesmaximal, complemented
3#23757193C212yesmaximal, complemented
4#23757190C212yesmaximal, complemented
5#23757152C212yesmaximal, complemented
6#23757144C214yesmaximal, complemented
7#23757136C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C157C211yesmaximal, characteristic, complemented
9C2×#14578954—24no—
10C2×#14578954—28no—
11C22×Dic314C2211yescharacteristic
12C22×D628C2211yescharacteristic
13C22×C22⋊D157C2212yes—
14C22×C4×D157C2212yes—
15#19097754—28no—
16C2×#14578933C2214yes—
17C2×C22⋊Dic157C2212yes—
18C2×#14578931C2214yes—
19C2×#14578931C2214yes—
20#19097678C2212yes—
21#19097678C2212yes—
22#19097678C2214yes—
23C23×C4×C157C2211yescharacteristic
24C23×C4×C157C2211yescharacteristic
25C23×C4×C157C2211yescharacteristic
26C2×C4⋊C4×C157—24no—
27C2×C4⋊C4×C157C2212yes—
28C2×C4⋊C4×C157C2212yes—
29C23×D157C2312yes—
30#14578954—232no—
31C2×Dic314—24no—
32C2×Dic314—22no—
33C2×D628—24no—
34C2×D628—22no—
35C2×C22⋊D157—24no—
36C2×C22⋊D157—24no—
37C2×C22⋊D157D418yes—
38C22×Dic157C2314yes—
39C22×Dic157C2312yes—
40C2×C4×D157—24no—
41C2×C4×D157—24no—
42C2×C4×D157—24no—
43#14578933—28no—
44C22⋊Dic157—24no—
45#14578931—28no—
46#14578931—28no—
47C24×C157C2311yescharacteristic
48C22×C4×C157—22no—
49C22×C4×C157—24no—
50C22×C4×C157—24no—
51C22×C4×C157—22no—
52C22×C4×C157—24no—
53C22×C4×C157D414yes—
54C22×C4×C157C2312yes—
55C22×C4×C157C2311yescharacteristic
56C22×C4×C157C2311yescharacteristic
57C22×C4×C157C2312yes—
58C4⋊C4×C157—24no—
59C4⋊C4×C157—24no—
60C22×D157—22no—
61C22×D157—22no—
62C22×D157—24no—
63C22×D157—24no—
64C22×D157C2×D414yes—
65Dic314—28no—
66D628—28no—
67C22⋊D157—216no—
68C2×Dic157—22no—
69C2×Dic157—22no—
70C2×Dic157—24no—
71C2×Dic157—24no—
72C2×Dic157—24no—
73C2×Dic157C2×D414yes—
74C4×D157—216no—
75C23×C157—22no—
76C23×C157—22no—
77C23×C157C2×D414yes—
78C23×C157C2×D412yes—
79C23×C157C2411yescharacteristic
80C2×C4×C157—28no—
81C2×C4×C157—28no—
82C2×C4×C157—22no—
83C2×C4×C157—22no—
84C2×C4×C157—22no—
85C2×C4×C157—22no—
86C2×C4×C157—22no—
87C2×C4×C157—24no—
88C2×C4×C157—44no—
89C2×C4×C157C2×D412yes—
90C2×C4×C157C2×D412yes—
91C2×C4×C157#4114yescomplemented
92C22×C157—22no—
93C22×C157—22no—
94C22×C157—28no—
95C22×C157#12014yes—
96C22×C157C22×D412yes—
97C22×C157C22×D411yescharacteristic
98C22×C157C2×#4112yes—
99C22×C1572−1+412yes—
100C2×D157—24no—
101C2×D157—28no—
102C2×D157—42no—
103C2×D157—42no—
104C2×D157—44no—
105C4×C157—22no—
106C4×C157—22no—
107C4×C157—24no—
108C4×C157—42no—
109Dic157—24no—
110Dic157—44no—
111C2×C157—24no—
112C2×C157C2×#12011yescharacteristic
113C2×C157#52511yescharacteristic
114C2×C157#53611yescharacteristic
115C2×C157#54714yes—
116D157—44no—
117C157#240611yescharacteristic, complemented, Sylow
118#2406—1571nomaximal, Sylow
119C22×#41—1571no—
120C2×#123—1572no—
121C22×C4⋊C4D15711yescharacteristic, complemented
122#395—1572no—
123#394—1572no—
124#386—1572no—
125#385—1574no—
126#381—1571no—
127C2×#41—3148no—
128C2×#41—3144no—
129C22×Q8—1571no—
130C22×D4—1572no—
131C22×D4—1571no—
132C23×C4—1572no—
133C23×C4C2×D15711yescharacteristic
134C23×C4C2×D15711yescharacteristic
135C23×C4C2×D15711yescharacteristic
136#123—3148no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—1574no—
139C2×C4⋊C4C2×D15712yes—
140C2×C4⋊C4C2×D15712yes—
141C2×C22⋊C4—1572no—
142C2×C22⋊C4—1574no—
143C2×C22⋊C4—1574no—
144#95—1572no—
145#95—1572no—
146#95—1574no—
147C24—1572no—
148C24C22×D15711yescharacteristic
149#41—31432no—
150C2×Q8—3144no—
151C2×Q8—3142no—
152C2×D4—3144no—
153C2×D4—3144no—
154C2×D4—3144no—
155C2×D4—3142no—
156C2×D4—1578no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—3144no—
163C22×C4—3144no—
164C22×C4—3144no—
165C22×C4—1574no—
166C22×C4—1572no—
167C22×C4C22⋊D15714yes—
168C22×C4C22×D15712yes—
169C22×C4C22×D15711yescharacteristic
170C22×C4C22×D15711yescharacteristic
171C22×C4C22×D15712yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—3148no—
175C22⋊C4—3148no—
176C22⋊C4—3144no—
177C22⋊C4—3148no—
178C23—22no—
179C23—22no—
180C23—3142no—
181C23—3142no—
182C23—3144no—
183C23—3144no—
184C23—1574no—
185C23D4×D15714yes—
186C23C2×C22⋊D15712yes—
187C23C23×D15711yescharacteristic
188Q8—3148no—
189D4—3148no—
190D4—31416no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—31416no—
201C2×C4—3142no—
202C2×C4—3142no—
203C2×C4—3144no—
204C2×C4—3144no—
205C2×C4—3144no—
206C2×C4—1574no—
207C2×C4C2×C22⋊D15712yes—
208C2×C4C2×C22⋊D15712yes—
209C2×C4#1457895414yescomplemented
210C22—22no—
211C22—28no—
212C22—22no—
213C22—3148no—
214C22—3144no—
215C22—6282no—
216C22—6284no—
217C22—6282no—
218C22#1909773914yes—
219C22C2×D4×D15712yes—
220C22C22×C22⋊D15711yescharacteristic
221C22C2×#1457895412yes—
222C22#1909781912yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—3144no—
228C4—6284no—
229C2—24no—
230C2—6284no—
231C2C2×#1909773911yescharacteristic
232C2#2375778411yescharacteristic
233C2#2375784711yescharacteristic
234C2#2375792414yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
FittingC22×C4⋊C4×C157C2
prime radicalC22×#14578954C2
hypercenterC22×C4⋊C4D157
hypocenterC157#2406
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#19097754C2semidirect
3. #23757193C2semidirect
4. #23757190C2semidirect
5. #23757152C2semidirect
6. #23757144C2semidirect
7. #23757136C2semidirect
8. C22×C4⋊C4×C157C2semidirect
91. C2×C4×C157#41semidirect
117. C157#2406semidirect
121. C22×C4⋊C4D157semidirect
209. C2×C4#14578954semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145789542
C2×#190977542
#237571932
#237571902
#237571522
#237571442
#237571362
C22×C4⋊C4×C1572
#2406157